Signal Sampling and Quantization - Digital Signal Processing: Fundamentals and Applications - page 27

frequency of f c in Hz, and a capacitor value of C 2 , we can determine the other elements using the
formulas listed in the figure.

1
R 1 R 2 C 1 C 2

2

ð 2 pf c Þ

1
R 1 C 2 þ

¼ (2.10)

2

1
R 2 C 2

1
R 1 R 2 C 1 C 2

2

s þ 1 : 4141 ð 2 pf c Þs þð 2 pf c Þ

s

2

þ s þ

As an example, for a cutoff frequency of 3,400 Hz, and by selecting C 2 ¼ 0 : 01 microfarad ( mF ),
we get

R 1 ¼ R 2 ¼ 6 ; 620 U; and C 1 ¼ 0 : 005 mF
Figure 2.18 shows the magnitude frequency response, where the absolute gain of the filter is
plotted. As we can see, the absolute attenuation begins at the level of 0.7 at 3,400 Hz and reduces
to 0.3 at 6,000 Hz. Ideally, we want the gain attenuation to be zero after 4,000 Hz if our sampling
rate is 8,000 Hz. Practically speaking, aliasing will occur anyway with some degree. We will study
achieving the higher-order analog filter via Butterworth and Chebyshev prototype function tables
in Chapter 8. More details of the circuit realization for the analog filter can be found in Chen
(1986).

1.1

1

0.9

0.8

0.7

0.6

0.5

f c =3400 Hz

0.4

0.3

0.2

0.1

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Frequency (Hz)

FIGURE 2.18
Magnitude frequency response of the second-order Butterworth lowpass filter.